Contents vii
20.5 Complex equations
217
20.6 The polar form of a complex number
218
20.7 Multiplication and division in polar form
220
20.8 Applications of complex numbers
221
21 De Moivre’s theorem
225
21.1 Introduction
225
21.2 Powers of complex numbers
225
21.3 Roots of complex numbers
226
21.4 The exponential form of a complex
number
228
22 The theory of matrices and determinants
231
22.1 Matrix notation
231
22.2 Addition, subtraction and multiplication
of matrices
231
22.3 The unit matrix
235
22.4 The determinant of a 2 by 2 matrix
235
22.5 The inverse or reciprocal of a 2 by 2 matrix 236
22.6 The determinant of a 3 by 3 matrix
237
22.7 The inverse or reciprocal of a 3 by 3 matrix 239
23 The solution of simultaneous equations by
matrices and determinants
241
23.1 Solution of simultaneous equations by
matrices
241
23.2 Solution of simultaneous equations by
determinants
243
23.3 Solution of simultaneous equations using
Cramers rule
247
23.4 Solution of simultaneous equations using
the Gaussian elimination method
248
Revision Test 7
250
24 Vectors
251
24.1 Introduction
251
24.2 Scalars and vectors
251
24.3 Drawing a vector
251
24.4 Addition of vectors by drawing
252
24.5 Resolving vectors into horizontal and
vertical components
254
24.6 Addition of vectors by calculation
255
24.7 Vector subtraction
260
24.8 Relative velocity
262
24.9 i, j and k notation
263
25 Methods of adding alternating waveforms
265
25.1 Combination of two periodic functions
265
25.2 Plotting periodic functions
265
25.3 Determining resultant phasors by drawing 267
25.4 Determining resultant phasors by the sine
and cosine rules
268
25.5 Determining resultant phasors by
horizontal and vertical components
270
25.6 Determining resultant phasors by complex
numbers
272
26 Scalar and vector products
275
26.1 The unit triad
275
26.2 The scalar product of two vectors
276
26.3 Vector products
280
26.4 Vector equation of a line
283
Revision Test 8
286
27 Methods of differentiation
287
27.1 Introduction to calculus
287
27.2 The gradient of a curve
287
27.3 Differentiation from first principles
288
27.4 Differentiation of common functions
289
27.5 Differentiation of a product
292
27.6 Differentiation of a quotient
293
27.7 Function of a function
295
27.8 Successive differentiation
296
28 Some applications of differentiation
299
28.1 Rates of change
299
28.2 Velocity and acceleration
300
28.3 Turning points
303
28.4 Practical problems involving maximum
and minimum values
307
28.5 Tangents and normals
311
28.6 Small changes
312
29 Differentiation of parametric equations
315
29.1 Introduction to parametric equations
315
29.2 Some common parametric equations
315
29.3 Differentiation in parameters
315
29.4 Further worked problems on
differentiation of parametric equations
318
30 Differentiation of implicit functions
320
30.1 Implicit functions
320
30.2 Differentiating implicit functions
320
30.3 Differentiating implicit functions
containing products and quotients
321
30.4 Further implicit differentiation
322
31 Logarithmic differentiation
325
31.1 Introduction to logarithmic differentiation 325
31.2 Laws of logarithms
325
31.3 Differentiation of logarithmic functions
325
20.5 Complex equations
217
20.6 The polar form of a complex number
218
20.7 Multiplication and division in polar form
220
20.8 Applications of complex numbers
221
21 De Moivre’s theorem
225
21.1 Introduction
225
21.2 Powers of complex numbers
225
21.3 Roots of complex numbers
226
21.4 The exponential form of a complex
number
228
22 The theory of matrices and determinants
231
22.1 Matrix notation
231
22.2 Addition, subtraction and multiplication
of matrices
231
22.3 The unit matrix
235
22.4 The determinant of a 2 by 2 matrix
235
22.5 The inverse or reciprocal of a 2 by 2 matrix 236
22.6 The determinant of a 3 by 3 matrix
237
22.7 The inverse or reciprocal of a 3 by 3 matrix 239
23 The solution of simultaneous equations by
matrices and determinants
241
23.1 Solution of simultaneous equations by
matrices
241
23.2 Solution of simultaneous equations by
determinants
243
23.3 Solution of simultaneous equations using
Cramers rule
247
23.4 Solution of simultaneous equations using
the Gaussian elimination method
248
Revision Test 7
250
24 Vectors
251
24.1 Introduction
251
24.2 Scalars and vectors
251
24.3 Drawing a vector
251
24.4 Addition of vectors by drawing
252
24.5 Resolving vectors into horizontal and
vertical components
254
24.6 Addition of vectors by calculation
255
24.7 Vector subtraction
260
24.8 Relative velocity
262
24.9 i, j and k notation
263
25 Methods of adding alternating waveforms
265
25.1 Combination of two periodic functions
265
25.2 Plotting periodic functions
265
25.3 Determining resultant phasors by drawing 267
25.4 Determining resultant phasors by the sine
and cosine rules
268
25.5 Determining resultant phasors by
horizontal and vertical components
270
25.6 Determining resultant phasors by complex
numbers
272
26 Scalar and vector products
275
26.1 The unit triad
275
26.2 The scalar product of two vectors
276
26.3 Vector products
280
26.4 Vector equation of a line
283
Revision Test 8
286
27 Methods of differentiation
287
27.1 Introduction to calculus
287
27.2 The gradient of a curve
287
27.3 Differentiation from first principles
288
27.4 Differentiation of common functions
289
27.5 Differentiation of a product
292
27.6 Differentiation of a quotient
293
27.7 Function of a function
295
27.8 Successive differentiation
296
28 Some applications of differentiation
299
28.1 Rates of change
299
28.2 Velocity and acceleration
300
28.3 Turning points
303
28.4 Practical problems involving maximum
and minimum values
307
28.5 Tangents and normals
311
28.6 Small changes
312
29 Differentiation of parametric equations
315
29.1 Introduction to parametric equations
315
29.2 Some common parametric equations
315
29.3 Differentiation in parameters
315
29.4 Further worked problems on
differentiation of parametric equations
318
30 Differentiation of implicit functions
320
30.1 Implicit functions
320
30.2 Differentiating implicit functions
320
30.3 Differentiating implicit functions
containing products and quotients
321
30.4 Further implicit differentiation
322
31 Logarithmic differentiation
325
31.1 Introduction to logarithmic differentiation 325
31.2 Laws of logarithms
325
31.3 Differentiation of logarithmic functions
325